The Straight Line: Know and Use the Definition of a Gradient

Understanding the Concept of Gradient

The gradient is a fundamental concept in coordinate geometry. It describes the steepness or inclination of a line.
In the context of a straight line, the gradient can be defined as the ‘rise’ divided by the ‘run’, in other words, the change in the vertical distance (y-values or height) divided by the change in horizontal distance (x-values or length).
- The ‘rise’ refers to the change in the y-coordinate (vertical change).
- The ‘run’ refers to the change in the x-coordinate (horizontal change).

The gradient (m) of a line can be calculated using the formula: m = (y2 - y1) / (x2 - x1)
Here, (x1, y1) and (x2, y2) represent any two points on the line.
Remember that the order of subtraction is very important. If you choose (x1, y1) as your first point, you must subtract it from the second point (x2, y2), not the other way round.

A positive gradient means that the line slopes upwards from left to right. A negative gradient implies that the line slopes downwards from left to right.
A line with a gradient of zero is a horizontal line. A vertical line has an undefined or infinite gradient.
The steeper the line, the larger the absolute value of the gradient. A steeper climb will result in a larger positive gradient, while a steeper fall will result in a larger negative gradient.

The numeric value of the gradient provides the ratio of ‘rise’ to ‘run’. If the gradient is 2 for instance, it implies that for every unit increase in x, the value of y increases by 2 units.
In the context of a graphical representation, if the gradient is -3, it implies that for every unit increase in x, the value of y decreases by 3 units.

Remember to always include the correct units when interpreting gradients in real world applications.